Community detection by simulated bifurcation

TL;DR

The paper addresses modularity-based community detection, an NP-hard problem, by formulating it as a QUBO/Ising optimization and solving with Simulated Bifurcation (SB), a quantum-inspired GPU algorithm. Using a one-hot community encoding with slack variables to enforce constraints, SB, particularly the discrete variant, efficiently explores the large solution space and yields high-modularity partitions. On Zachary's Karate Club, SB achieves $Q_e=0.445$ at $K_{opt}=4$, and on the IEEE 33-Bus system, SB reaches $Q_e=0.743$ at $K_{opt}=7$, outperforming IBM, D-Wave, and Gurobi and matching Fujitsu's Digital Annealer. These results illustrate SB’s potential as a scalable, hardware-efficient alternative to quantum devices for hard combinatorial optimization in network analysis.

Abstract

Community detection, also known as graph partitioning, is a well-known NP-hard combinatorial optimization problem with applications in diverse fields such as complex network theory, transportation, and smart power grids. The problem's solution space grows drastically with the number of vertices and subgroups, making efficient algorithms crucial. In recent years, quantum computing has emerged as a promising approach to tackling NP-hard problems. This study explores the use of a quantum-inspired algorithm, Simulated Bifurcation (SB), for community detection. Modularity is employed as both the objective function and a metric to evaluate the solutions. The community detection problem is formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem, enabling seamless integration with the SB algorithm. Experimental results demonstrate that SB effectively identifies community structures in benchmark networks such as Zachary's Karate Club and the IEEE 33-bus system. Remarkably, SB achieved the highest modularity, matching the performance of Fujitsu's Digital Annealer, while surpassing results obtained from two quantum machines, D-Wave and IBM. These findings highlight the potential of Simulated Bifurcation as a powerful tool for solving community detection problems.

Figures (4)

• Figure 1: Modularity ($Q_e$) versus Community Number ($K$) for Zachary's Karate Club using Simulated Bifurcation. The optimal result is achieved at $K_{\mathrm{opt}} = 4$ with $Q_e = 0.445$.
• Figure 2: Partitioned Graph with the highest Modularity $Q_e=0.445$, with Community Number $K_{\mathrm{opt}}=4$ by Simulated Bifurcation for Zachary's Karate Club Network.
• Figure 3: Modularity ($Q_e$) versus Community Number ($K$) for the IEEE 33–Bus System using Simulated Bifurcation. The optimal result is achieved at $K_{\mathrm{opt}}=7$ with $Q_e=0.743$.
• Figure 4: Partitioned Graph with the highest Modularity $Q_e=0.743$, with Community Number $K_{\mathrm{opt}}=7$ by Simulated Bifurcation for IEEE 33-Bus System.